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Let's check that

$$ g(x) + h(x) = \lfloor x \rfloor $$

By definition

$$ g(x) + h(x) = m_x + \lfloor s_x + n_x \sqrt{2} \rfloor $$

but you can push an integer inside the floor function so

$$ g(x) + h(x) = \lfloor m_x + s_x + n_x \sqrt{2} \rfloor =\lfloor x \rfloor $$

If I understand correctly, $m_x$ doesn't change when you add $\sqrt{2}$ to $x$, $n_x$ is unchanged when you add $1$ to $x$, and $s_x$ is unchanged when you add either $\sqrt{2}$ or $1$ to $x$.